This paper presents a method to replace interaction in zero-knowledge proofs with a shared random string, enabling non-interactive zero-knowledge proofs. The authors show that this approach allows for the construction of the first public-key cryptosystem secure against chosen ciphertext attacks. They also demonstrate that non-interactive zero-knowledge proofs can be used to prove the validity of any theorem without revealing any information beyond the theorem's truth. The paper discusses the implications of this result for cryptography and complexity theory, including the construction of efficient identification schemes and the robustness of the result under various conditions. The authors also introduce a new complexity assumption that is sufficient for non-interactive zero-knowledge proofs and show how it can be applied to construct a single-theorem non-interactive zero-knowledge proof system for 3-colorable graphs. They further extend this result to non-interactive zero-knowledge proofs for multiple theorems and discuss the implications for public-key cryptography, including the construction of a cryptosystem secure against chosen ciphertext attacks. The paper concludes with a discussion of the broader implications of their results for cryptography and complexity theory.This paper presents a method to replace interaction in zero-knowledge proofs with a shared random string, enabling non-interactive zero-knowledge proofs. The authors show that this approach allows for the construction of the first public-key cryptosystem secure against chosen ciphertext attacks. They also demonstrate that non-interactive zero-knowledge proofs can be used to prove the validity of any theorem without revealing any information beyond the theorem's truth. The paper discusses the implications of this result for cryptography and complexity theory, including the construction of efficient identification schemes and the robustness of the result under various conditions. The authors also introduce a new complexity assumption that is sufficient for non-interactive zero-knowledge proofs and show how it can be applied to construct a single-theorem non-interactive zero-knowledge proof system for 3-colorable graphs. They further extend this result to non-interactive zero-knowledge proofs for multiple theorems and discuss the implications for public-key cryptography, including the construction of a cryptosystem secure against chosen ciphertext attacks. The paper concludes with a discussion of the broader implications of their results for cryptography and complexity theory.